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A new hybrid technique of cuckoo search and harmony search for solving non-smooth optimal
power flow framework
Aboubakr Khelifi*, Saliha Chettih, Bachir Bentouati
University of Ammar Teledji Laghouat, Electrical Engineering Department, Laghouat, Algeria
Corresponding Author Email: khelifi@lagh-univ.dz
https://doi.org/10.18280/ama_b.610402
Received: 15 March 2018
Accepted: 5 June 2018
ABSTRACT
In order to improve the search capability of the existing Cuckoo Search (CS) algorithm, an
enhanced robust technique is proposed in this paper, called hybrid Cuckoo Search and
Harmony Search (CSHS). In CSHS technique, HS incorporates the mutation operator into
the Cuckoo Search technique. The proposed technique is applied to solve the highly
nonlinear and non-convex optimal power flow (OPF) problem. In this paper, OPF is
mathematically formulated as nonlinear multi-objective optimization problem. The
developed formulation minimizes simultaneously the conflicting objectives of fuel cost,
valve-point effect, emission reduction, voltage profile improvement and voltage stability
enhancement subject to system equality and inequality constraints. OPF problem is solved
using the proposed CSHS algorithm and tested on standard IEEE 30-bus and IEEE 57-bus
with different case studies. The results obtained are compared with the reported literature.
The results demonstrate that the proposed algorithm outperforms the original CS and HS
and other algorithms available in the literature.
Keywords:
Cuckoo Search, harmony search, optimal
power flow, emission, constraints
1. INTRODUCTION
The Optimal Power Flow (OPF) is a significant appliance
for planning and operation studies in the power system
operator. OPF is a widely non-linear and non-convex
optimization problem, and this is more difficulty in practical
applications in the large number's presence of discrete
variables. The goal of OPF is to provide the optimal settings
of the power system by improving objective function while
meeting the equality and inequality constraints [1], then this
problem has been addressed by several researchers. The
objective functions, such as the minimization of total fuel
cost, improvement of voltage stability index and reduction of
real power loss are considered individually in the literature
for this study [2]. The problem of power flow is one of the
fundamental problems in which the load and the powers of
generator are given or corrected. The OPF has a long history
in its development, and it was primary introduced by
Carpentier in 1962 [1] and the next investigations on OPF in
[2]. However, it took a long time to turn into an effective
technique that could be applied in daily use. The actual
interest for OPF is focused on its capability to solve for the
optimal solution that has considered the security of the
system. The optimal power flow has been applied to regulate
the active power outputs and voltages of the generator,
transformer tap settings, shunt reactors/capacitors and other
controllable variables to minimize the generator fuel cost,
network active power loss, voltage stability index, while
keeping all the load bus voltages, generator reactive power
outputs, network power flows, and all other state variables in
the power system within their secure and operational bounds.
In its most common problem formulation, the OPF is a non-
convex, static, large-scale optimization problem with both
continuous and discrete control variables [3]. Even in the
absence of non-convex generator operating cost functions,
prohibited operating zones (POZs) of generating units, and
discrete control variables, the OPF problem is a non-convex
because of the existence of the non-linear alternating current
power flow equality constraints. The existence of discrete
control variables, such as transformer tap positions,
switchable shunt devices, phase shifters, further complicates
the formulation and solution of the problem [4].
Different conventional optimization methods have been
used to solve the OPF problem. These involve newton
methods [5], interior point method [6], and linear
programming [7], a comprehensive survey of different
conventional optimization techniques used to solve OPF
problems was presented. Nevertheless, in practice,
conventional techniques suffer from some weakness. Some
of its shortcomings through other things are: First, they do
not assure to find the global optimum, second, conventional
techniques involve complex computations with a long time,
and they do not suitable for discrete variables [8].
During the last little decades, a lot powerful meta-
heuristics were developed. Several of them have been
implemented to the OPF problem with very successfully.
various of the modern implementations of meta-heuristics for
the OPF problem are: Particle Swarm Optimization (PSO)
[9,10], Hopfield Neural Network (HNN) [11], Elephant
Herding Optimization (EHO) [12], Moth Flam Optimizer
(MFO) [13], Biogeography-Based Optimization (BBO) [14],
Differential Search Algorithm (DSA) [15], Multi-Verse
Optimizer (MVO) [16], and Krill Herd Algorithm (KHA)
[17]. Though, because of changing objectives while solving
OPF problems, no algorithm is the greatest one to solve all
the OPF problems. Consequently, there is still a need for a
novel algorithm, which can effectively solve the most of OPF
problems. In addition, enhanced efficiency is often carried
Advances in Modelling and Analysis B Vol. 61, No. 4, December, 2018, pp. 176-188
Journal homepage: http://iieta.org/Journals/AMA/AMA_B
176
out by hybridizing the technique and deterministic
optimization methods. Enhancing the searching capability of
the optimal solutions is the goal of technique hybridization.
The Cuckoo Search (CS) technique is a perfectly new
optimization algorithm, which is designed based on the Lévy
flight and brood parasitic behavior of certain cuckoo species
[18]. Further, CS can yield optimum solution but the
searching operation using levy flight cannot be assured. So as
to overcome the problem, Harmony Search (HS) [19] can be
one of the method to be incorporated with cuckoo search. HS
can give a mutation operator to the Cuckoo Search technique.
Thus, the exploitation ability of the solution will be best. By
using the characteristics of CS and HS, this paper suggests a
hybrid (CSHS), The effectiveness of this technique is utilized
to keep away from local optima and get a worldwide ideal
solution, in addition, less computational time to achieve the
ideal solution, local minima evasion, and speeder
convergence, which make them adequate for viable
applications for solving various constrained optimization
problems. The goal of this paper is to develop an enhanced
CS called CSHS to solve OPF problem. In order to justify the
development of the CSHS, its efficiencies are compared to
CS, HS and other well-known optimization techniques. Two
exam systems networks IEEE 30-bus and IEEE 57-bus test
systems are considered.
The remainder of paper is organized in the following way:
The next partout lines the formulation of the OPF problem;
meanwhile, section 3 describes the algebraic equation of
CSHS. Section 4 displays the results of simulation and
discussion. Finally the conclusion of this work is in section 5.
2. OPTIMAL POWER FLOW (OPF)
The OPF is a power flow problem that provides the
optimal settings of the control variables for specific settings
of load by means of reducing a predefined objective function
such as the cost of real power generation or transmission
losses. OPF takes into account the operating limits of the
system and it can be mathematically formulated as a
nonlinear constrained optimization problem as follows:
Minimize: J (𝑥, 𝑢) (1)
( ), 0g x u
Subject to:
( ), 0h x u =
where J (x,u), objective function; h (x,u), set of equality
constraints; g(x,u), set of inequality constraints; u vector of
control variables; x, vector of state variables; The control
variables u and the state variables x of the OPF problem are
explained in relations (2) and (3), respectively.
Control variables:
These are the set of variables that can be regulated to
satisfy the load flow equations [20]. The set of control
variables in the mathematical formulation of the OPF
problem are:
PG: is the 𝑖-th active power bus generator (except swing
generator).VG: is the voltage magnitude at 𝑖-th PV bus
(generator bus). T: is a transformer tap setting.
QC: is shunt VAR compensation.
The control variables U can be formulated as:
2 1 1 1... , ... , ... , ...NG NG NCG G G G C C NTu P P V V Q Q T T = (2)
where NC, NT and NG are the number of VAR compensators,
the number of regulating transformers and the number of
generators respectively.
State variables:
These are the set of variables that report any unique state
of the system [20]. The set of state variables for
mathematically formulated the OPF problem as follow:
PG1: is the generator active power at slack (or swing) bus.
VL: is the bus voltage of 𝑝-th load bus (PQ bus).
QG: reactive power generation of all generator units.
SL: transmission line loading (or line flow)
The state variables X can be formulated as:
1 1 1 1, ... , ... , ...
NL NG nlG L L G G l lx P V V Q Q S S = (3)
where, NL, and nl are the number of load buses and the
number of transmission lines, respectively.
Constraints:
OPF constraints can be classified into equality and
inequality constraints, as explained in the next sections.
Equality constraints
The equality constraints that express the typical nonlinear
power flow equations that control the power system,
presented as follows.
a) Real power constraints:
( ) ( )1
cos sin 0i i
NB
G D i j ij ij ij ij
j
P P V V G B =
− − + =
(4)
Reactive power constraints:
(5)
where NB is the number of buses, 𝑃𝐷 and 𝑄𝐷 are active and
reactive load demands, respectively, ij i j = − is the
difference in voltage angles between bus 𝑖 and bus 𝑗 ijG is
the transfer conductance and ijB is the susceptance between
bus 𝑖 and bus 𝑗, respectively.
Inequality constraints:
The Inequality constraints that reflect operational of the
system and the system's physical limits presented as follows.
Generator constraints. For all generators comprising the
slack: voltage, active and reactive outputs shall to be limited
by their minimum and maximum limits as follows:
min max
i i iG G GV V V i NG (6)
min max
i i iG G GP P P i NG (7)
min max
i i iG G GQ Q Q i NG (8)
( ) ( )1
sin cos 0i i
NB
G D i j ij ij ij ij
j
Q Q V V G B =
− − + =
177
Transformer constraints. Transformer tap settings must be
limited within their specified minimum and maximum limits
as follows:
min max
j j jT T T j NT (9)
Shunt VAR compensator constraints. Shunt VAR
compensators have to be limited by their lower and upper
limits as follows:
min max
k k kC C CQ Q Q k NC (10)
Security constraints:
These comprise the constraints of voltage magnitude at
load buses and transmission line loadings. Voltage of each
load bus has to be limited within its minimum and maximum
operating limits. Line flow through each transmission line
must be limited by its capacity limits. These constraints can
be expressed as given follows:
min max
p p pL L LV V V p NL (11)
max
q ql lS S q nl (12)
where
min
pLV and
max
pLVrepresents lowest and upper load
voltage of ith unit, qlS
represents apparent power flow of ith
branch
max
qlS
represents maximum apparent power flow limit
of ith branch.
3. HARMONY SEARCH
In 2001, Geem et al. first proposed the harmony search
(HS) algorithm [19], the fundamental HS technique involves
the following operators: the harmony memory (HM) [see Eq.
(13)], the harmony memory size (HMS), the harmony
memory consideration rate (HMCR), the pitch adjustment
rate (PAR) and the pitch adjustment bandwidth (bw ).
( )
( )
( )
1 1 1 1
1 2
2 2 2 2
1 2
1 2
... fitness
... fitness
...
... fitness
D
D
HMS HMS HMS HMS
D
x x x x
x x x xHM
x x x x
= (13)
Within HS, there are three elements: use of harmony
memory, pitch adjusting, and randomization. In the process
of HS optimization, the value of each decision variable in
HM can be specified by one of the previously mentioned
three rules.
The first section is important in the entire HS process. This
can assure that the preferable harmonies cannot be varied and
make the HM always stay the preferable status. HMCR ∈ [0,
1] must be cautiously adjusted with the goal of using this
memory more successfully. If it is nears 1 (very high), almost
whole the harmonies in them can be completely exploited,
but the HS algorithm cannot perform a global search, leading
to possible wrong solutions. On the other hand, if it is too
small (even 0), HS uses only a few preferable harmonies,
which may result to slowly finding the preferable solutions.
Here, usually, HMCR = 0.7-0.95. For the second section,
although the pitch can be lightly adjusted in the linear form
or nonlinear theoretically, a linear adjustment is selection in
most cases. The pitch is modernized as follows:
( )2 1new oldx x bw = + −
(14)
where δ is a number's random in [0, 1], bw is the band
width. oldx And newx
are the actual and novel pitches,
respectively.
Pitch adjustment has the likeness with the mutation
operator in evolutionary techniques. Likewise, the PAR is
also cautiously adjusted in order to implement HS in the best
way. If PAR gets closer to 1, the harmony in HM will switch
even at the finale of the HS operation, and HS is therefore
difficult to converge on the best solutions. Conversely, if it is
too low, then a slight change will be made for harmonies in
HS and HM might converge prematurely. Hence, we use
PAR = 0.1–0.5 for generality cases. The third section is
basically a random process with the goal of adding harmony
diversity. The random operation makes the HS explore the
entire search space excellently and this has a greater prospect
of finding the final optimal solutions.
4. CUCKOO SEARCH
By simplifying and perfecting the parasitic the conduct of
the brood of cuckoos individuals in incorporation with Lévy
flight, CS is proposed which is a new technique of meta-
heuristic research [18] to solve optimization problems.
In the state of CS, how a cuckoo individual moves to the
following position is entirely specified by Lévy flights.
To use cuckoo brood the conduct to optimization problems,
Yang and Dib are ideal for the brood parasitic the conduct of
some cuckoo, the following three rules have been developed
forward.
1. In the cuckoo population, every cuckoo puts an egg at a
nest chosen at random.
2. Great -quality nests will not be changed, and this can
assure the cuckoo population that involves the superior
solutions, not worse than previously at least.
3. The number of nests is not changed and the egg laid by
a cuckoo is found by the host bird with a possibility pa ∈ [0,
1].
In the easy form, every nest only repays only to one an egg.
As a result, the CS technique can be simply extended to
address multi-objective optimization problems in which
every nest comprises more than one egg / solution. In our
current study, we only consider that every nest has just an
egg. So, in our study, we do not determine the difference
between the nest, egg, and solution. CS technique can
achieve a perfect balance between the local random walk and
the global random walk by utilizing a pa switch parameter.
The local one can be expressed as [18]:
178
( ) ( )1t t t t
i i s a j kX X H p X X + = + − − (15)
Where 𝑋𝑗𝑡 and 𝑋𝑘
𝑡 are two diverse solutions randomly chosen,
H(u) is a Heaviside function, ε is a number's random, and s is
the step size, For the global random walk, it is incorporated
with Lévy flights given as follows:
( ) ( )( )
( )1
01
sin12
, , , , , 0t t
i iX X L s L s s ss
+
= + =
+ (16)
Here 𝛽 ≻ 0 is the scaling factor of the step size concerning
to the scales of the interest problem.
5. HYBRID HARMONY SEARCH AND CUCKOO
SEARCH
Based on the introduction of CS and HS in earlier section,
the detailed characterize of the suggested cuckoo search with
harmony search (HS/CS) will be presented in this section.
In general, the standard CS technique explores the search
space well and has a quick speed to find the optimal global
value, but it takes exploits of solutions badly because of the
moves or sometimes large steps. Furthermore, standard
harmony search is well able to exploit solutions by carefully
adjusting the HMCR and PAR. In the display work, by
combination of HS and CS, a hybrid meta-heuristic technique,
so-called Cuckoo Search/harmony Search (CS/HS) is thus
suggested for the goal of optimizing benchmark functions. In
CS/HS technique, the improvisation of harmony in HS is
inserting into cuckoo search as operator of mutation. In this
process, this technique can explore the modern search space
by hybrid CS operator and exploit population with HS, and
thus, the benefits of the CS and HS can be fully utilized.
The basic idea of the HS/CS technique is the provided of
the hybrid HS mutation operator. In this way, first introduced
in the current work, a major improvement is made to add a
mutation operator to the CS including two minor
improvements.
The first improvement is the addition of the pitch
adjustment process in the HS, which can be considered a
mutation operator in order to augment the diversity of the
population. In the exploitation phase, once an individual is
selected among the best current individuals, a new Cuckoo
individual is created globally using Lévy flights. After that,
we adjust each element in 𝑥𝑖 using HS. When 𝜉 is greater
than HMCR, i.e., 𝜉1 ≥ HMCR, the component j is updated
randomly; whereas when 𝜉1< HMCR, we update component j
according to 𝑥∗, furthermore, pitch adjustment process in HS
which can be considered as a mutation operator is applied to
update the component j when 𝜉2<PAR in a purpose to add
diversity of the population, as described in equation (14),
where 𝜉1 and 𝜉2 are two random numbers distributed
uniformly in [0,1], 𝑥∗ is the global preferable solution in the
current generation. By means of different experiments, it was
found that HMCR is specific to 0.9 and PAR to 0.1 which
can produce optimal solutions.
The else improvement is to add of elitist scheme to into the
HS/CS.As with else optimization techniques, an improved
elitism scheme is combined into the HS/CS algorithm to
retain the preferable individuals in the cuckoo population.
According to the above detailing, the harmony
search/cuckoo search (HS/CS) can be found in the
corresponding flowchart appears in Figure 1.
Figure 1. The flowchart of Hybrid CSHS algorithm
6. APPLICATION AND RESULTS
The CSHS has been using to solve the OPF problem for
two exam systems and for many cases with various objective
functions. The considered power systems networks are: the
IEEE 30-bus and IEEE 57-bus test system. The advanced
software program is written in MATLAB computing
environment and used on a 2.20 GHz i7 personal computer.
In our study the CSHS population size or number of stars is
selection to be 50.
6.1 IEEE 30-bus test system
In order to illustrate the performance of the proposed
CSHS method, it has been examined first on the standard
IEEE 30-bus test system. The standard IEEE 30-bus system
selection in this paper has the next characteristics: 6-
generators at buses 1, 2, 5, 8, 11 and 13, 4-transformers with
off-nominal tap ratio at lines 11, 12, 15 and 36, 9- shunt VAR
compensation buses at buses 10, 12, 15, 17, 20, 21, 23, 24
and 29. In addition, line data, bus data, generator data, and
lower and upper restriction for control variables are presented
in [21]. For this first exam system, six various cases have
179
been studied with various objectives and all the obtained
results are outlined in Table 1, 3 and 5. The first column of
this table appears the optimal control settings found where:
- PG1 through PG6 and VG1 through VG6 represent the
powers and the voltages of generator 1 through generator 6.
- T11, T12, T15 and T36 are the tap settings of transforms
involved between lines 11, 12, 15 and 36.
- QC10, QC12, QC15, QC17, QC20, QC21, QC23, QC24
and QC29 represent the shunt VAR compensations
connected to buses 10, 12, 15, 17, 20, 21, 23, 24 and
29.moreover, fuel cost ($/h), active power losses (MW),
reactive power losses (MVar), voltage deviation and Lmax
represent the total fuel cost of the system, the total active
transmission losses, the deviation of load voltages and the
index of stability, respectively. More description about these
results will be presented in the next sections.
Case 1: Minimization of generation fuel cost
The first case studied in this article is the basic case of
minimizing the cost generation fuel expressed by a quadratic
function. Therefore, the objective function of this case is:
( )1
$ /NG
i
i
J f h=
= (17)
where fi is the fuel cost of the ith generator. Usually, the OPF
generation fuel cost curve is formulated by a quadratic
function. Hence, fi can be formulated as follows [16]:
( )2
i ii i i G i Gf a b P c P= + + (18)
where 𝑎𝑖 , 𝑏𝑖 and 𝑐𝑖 are the element, the linear and the
quadratic cost coefficients of the ith generator, respectively.
The values of these coefficients are presented in [21].
Figure 2 appears the trend of total fuel cost over iterations.
It seems that the proposed technique has good convergence
characteristics. The optimal settings of control variables are
presented in table 1. The total fuel cost obtained by the
suggested CSHS technique is (798.9166$/h). Compared to
the original CS, HS the total fuel cost is significantly
decreased. Using the identical conditions (limits of control
variables, initial conditions, and system data), the results
obtained in Case 1 apply the CSHS technique are compared
to other methods described in the literature as appears in
Table 2. There is some proof, that the suggested technique
outperforms several methods used to solve the OPF problem
by decreasing of generation fuel cost. For instance, the results
obtained by the CSHS are better than the ones obtained the
CS and HS methods.
Figure 2. Objective function curve for CASE 1
Figure 3. Objective function curve for CASE 2
Table 1. Optimum control variables for case 1 and case 2
Case1 Case2
Control variable CS-HS CS HS CS-HS CS HS
PG1 (MW) 177.1113 177.1178 178.1042 200.0281 200.0828
200.0000
PG2 (MW) 48.6899 48.6915 49.1159 42.8209 41.8634
43.5433
PG5 (MW) 21.303 21.3039 21.3845 18.8234 18.4909 18.6056
PG8 (MW) 21.0241 21.0311 21.6787 10.0000 11.2835
0 50 100 150 200 250 300 350 400 450 500798
799.5
801
802.5
804
805.5
807
805.5
810
811.5
Iteration
Fu
el
co
st
($/h
)
CS
HS
CS-HS
0 50 100 150 200 250 300 350 400 450 500830
835
840
845
850
855
860
865
Iteration
Fu
elco
st (
$/h
)
CS
HS
CS-HS
180
10.0000
PG11 (MW) 11.8572 11.8567 10 10.0033 10.0112
10.0000
PG13 (MW) 12 12 12 12.0129 12.0137
12.0000
V1(p.u) 1.1 1.1 1.1 1.1000 1.1000
1.1000
V2(p.u) 1.08768 1.08769 1.1 1.0868 1.0859
1.1000
V5(p.u) 1.06133 1.06131 1.07287 1.0589 1.0604 1.0811
V8(p.u) 1.06906 1.06911 1.07947 1.0649 1.0685
1.1000
V11(p.u) 1.1 1.1 1.1 1.0994 1.0982
1.1000
V13(p.u) 1.1 1.1 1.1 1.0999 1.0972
1.1000
Qc10(Mvar) 5 5 5 1.9804 0.1835
0
Qc12(Mvar) 5 5 5 0.2174 0.2306
0
Qc15(Mvar) 5 5 5 2.0893 0.1200
3.7031
Qc17(Mvar) 5 5 5 0.8167 0.2224 0
Qc20(Mvar) 5 0 0 2.9613 3.4344
0
Qc21(Mvar) 5 5 5 5.0000 2.6778 5.0000
Qc23(Mvar) 2.60333 3.23431 5 0.0205 0.0687 5.0000
Qc24(Mvar) 5 5 5 5.0000 1.4904 5.0000
Qc29(Mvar) 2.29632 2.37528 5 3.9313 1.6051 5.0000
T6–9 1.04067 1.036 0.969235 1.0262 1.0821 1.0058
T6–10 0.9 0.9 1.1 0.9225 0.9281 1.1000
T4–12 0.977254 0.984169 1.1 1.0336 1.0598 1.1000
T28–27 0.960932 0.964347 1.01459 0.9826 0.9895 1.1000
Fuel cost ($/h) 798.9166 798.9706 799.7727 829.9584 830.4784
831.5945
VD 1.9733 1.8751 1.4034 1.4402 0.9963
0.9863
maxL 0.1261 0.1269 0.1336 0.1316 0.1362
0.1456
Emission (ton/h) 0.3662 0.3662 0.3695 0.4426 0.4426
0.4425
( )lossp MW 8.5855 8.6010 8.8833 10.2887 10.3455 10.7490
Case 2: Minimization of fuel cost considering valve point
effect
So as to have a realistic and greater effective modeling of
generator cost functions, the valve point–effect must be
considered. The generating units with multi-valve steam
turbines display a major variation in the fuel-cost functions
and output a ripple-like effect. So as to considered the valve-
point effect of generating units, a modeled as a sinusoidal
term is added to the cost function [24]. Thus, the objective
function can be formulated as follow:
(20)
where, 𝑑𝑖and 𝑒𝑖are the coefficients that show the valve-point
loading effect.
In this case to arrive at a rise in cost than in case 1 with
conclusive value being 829.9584$/h, obtained by CSHS. The
optimal control variables obtained are shown in Table1,
output outcome of method used in our study are better than
most of the results revealed in past literatures on the problem
of OPF.
Table 2. Comparison of results for case1 and case 2
Case 1 Case 2
Algorithms Fuel
cost($/h) Algorithms Fuel cost($/h)
CS-HS 798.9166 CS-HS 829.9584
CS 798.9706 CS 830.4784
HS 799.7727 HS 831.5945
DE [21] 799.289 BSA [20] 830.7779
SOS[22] 801.5733 ICBO [25] 830.4531
MSFLA[23] 802.287 APFPA [26] 830.4065
HSFLA-
SA[24] 801.79
Case 3: Minimization of fuel cost and voltage deviation
Bus voltage is one of the most significant and considerable
security and service quality indices [21]. Reducing only the
total cost in the OPF problem as in Case 1 may result in a
suitable solution, but voltage profile may not be reasonable.
( ) ( )2 min
1
, ( ( )NG
i i i i i i i gi gi
i
f x u a b P c P d e P P=
= + + + −
181
Consequently, this case purposes at minimizing fuel cost with
a improve voltage profile by considering a dual objective
function. The voltage profile is optimized by reducing the
load bus voltage deviation (VD) from 1.0 p.u, the objective
function in this case can be formulated as follows:
cost voltageDeviationJ J wJ= + (21)
where w is an appropriate weighting factor, to be chosen by
the user to accord a weight to each of the two expressions of
the objective function. In this case w is selection as 100.
costJ and VoltageDeviationJ
are presented as follows:
cos
1
NG
t i
i
J f=
= (22)
1
1NL
voltageDeviation i
k
J V=
= − (23)
The CSHS technique has been utilized to search for the
optimal solution of the problem. The variations in the fuel
cost and voltage deviation through the iterations are outlined
in Fig.4a and Fig.4b. The optimal settings of the control
variables are presented in table 3. Apply CSHS the fuel cost
and the voltage deviation yielded are (803.5208$/h) and
(0.0991p.u), respectively. The voltage profile obtained by
CSHS is compared with other algorithms as appears in table
4. It is clear that the voltage profile is the least among all
other comparable methods. It is decreased from 1.9733p.u. In
the case 1 to 0.0991p.u in case 3, hence, in case 3, the fuel
cost is slightly augmented by 0.57% compared to case 1.
Figure 4a. Objective function curve for CASE3
Figure 4b. Objective function curve for CASE3
Table 3. Optimum control variables for case 3 and case 4
Case3 Case4
Control variable CS-HS CS HS CS-HS CS HS
PG1 (MW) 176.1521 177.8590 176.5090 178.0124 177.2886 177.8314
PG2 (MW) 48.6077 49.0616 48.9845 49.4689 50.1364 49.0497
PG5 (MW) 21.6759 21.6616 21.5998 21.2303 20.9317 21.4515
PG8 (MW) 22.5226 22.9236 22.3333 19.4564 18.0767 22.1245
PG11 (MW) 12.2148 10 11.9979 11.7191 12.5048 10.0000
PG13 (MW) 12 12 12 12.3024 13.3872 12.0000
V1(p.u) 1.03871 1.05487 1.04718 1.1000 1.1000 1.1000
V2(p.u) 1.02104 1.036 1.0312 1.0886 1.0905 1.1000
V5(p.u) 1.00946 1.01811 1.01833 1.0676 1.0667 1.1000
V8(p.u) 0.99976 1.00864 1.01295 1.0749 1.0727 1.1
V11(p.u) 1.07605 0.9969 0.95 1.1000 1.1000 1.1
V13(p.u) 0.99678 0.96439 0.98414 1.1000 1.1000 1.1
Qc10(Mvar) 4.9997 0 5 1.9688 0.5394 5
Qc12(Mvar) 0 0 5 0.4147 1.3942 0
Qc15(Mvar) 5 5 5 0 0.1355 5.0000
Qc17(Mvar) 0 3.65345 5 2.6176 0.5193 5.0000
0 50 100 150 200 250 300 350 400 450 500800
810
820
830
840
850
860
870
880
890
Iteration
Fu
el
co
st
($/h
)
CS
HS
CS-HS
0 50 100 150 200 250 300 350 400 450 500
0.16
0.22
Convergence curve
Iteration
Vol
tage
dev
iati
on (
p.u
)
CS
HS
CS-HS
182
Qc20(Mvar) 5 4.99998 5 3.4887 0.0944 0
Qc21(Mvar) 5 5 5 4.3462 0.1221 0
Qc23(Mvar) 5 5 5 1.7656 3.3164 4.3684
Qc24(Mvar) 5 5 0 5.0000 0.0099 5.0000
Qc29(Mvar) 2.49152 5 1.76278 0.2188 0.3625 2.1684
T6–9 1.1 1.00336 0.96716 1.0400 1.0075 1.1000
T6–10 0.9 0.9 0.9 0.9192 0.9043 0.9000
T4–12 0.95347 0.9 0.94311 0.9835 1.0060 0.9996
T28–27 0.96762 0.97936 0.95791 0.9416 0.9313 0.9723
Fuel cost ($/h) 803.5208 804.2983 804.2596 799.3251 800.0275 800.4916
VD 0.0991 0.1040 0.1122 1.7688 1.6361 1.9221
maxL 0.1487 0.1493 0.1482 0.1251 0.1254 0.1249
Emission (ton/h) 0.3632 0.3686 0.3643 0.3689 0.3666 0.3688
( )lossp MW 9.7731 10.1058 10.0245 8.7895 8.9254 9.0572
Table 4. Comparison of results for case 3 and case 4
Case 3 Case 4
Algorithms Fuel
cost($/h)
VD(p.u) Algorithms Fuel cost($/h) maxL
CS-HS 803.5208 0.0991 CS-HS 799.3251 0.1251
CS 804.2983 0.1040 CS 800.0275 0.1254
HS 804.2596 0.1122 HS 800.4916 0.1249
BSA [20] 803.4294 0.1147 ARCBBO
[27]
801.8076 0.1369
DE [21] 805.2620 0.1357 BSA [20] 800.3340 0.1259
BBO [14] 804.9980 0.102 ABC [28] 801.6650 0.1379
GABC[28] 801.5821 0.137
Figure 5a. Objective function curve for CASE 4
Figure 5b. Objective function curve for CASE 4
Case 4: Minimization of fuel cost and enhancement of
voltage stability
The prediction of voltage instability is a problem of
paramount significance in power systems. In [32] Kessel and
Glavitch have developed a voltage stability index named
which is defined build on local indicators and it is
presented by [32]:
(24)
where is the local indicator of bus j and it is given as
follows:
0 50 100 150 200 250 300 350 400 450 500795
805
815
825
835
845
855
Iteration
Fu
el c
ost
($/h
)
CS
HS
CS-HS
0 50 100 150 200 250 300 350 400 450 5000,1250
0,126
0,127
0,128
0,129
0,13
Iteration
Lm
ax
CS
HS
CS-HS
maxL jL
( )max max jL L= 1,2,...,j NL=
jL
183
(25)
where H matrix is produced by the partial inversion of .
More specifics can be given in [32]. The indicator
varies between 0 and 1 where the lower the indicator, the
more the system stable. Thus, enhancing voltage stability can
be obtained by the minimization of of the complete
system [21]. Hence, the objective function can be formulated
as:
(26)
where is a weighting factor chosen as 100 in this
work. The results of the optimization study are presented in
Table 3 while the direction of convergence is appearing in
Fig. 5. It seems that the has been decreased from 0.1283
to 0.1251 compared with CASE 1. Hence the results obtained
are compared with other algorithms as given in table 4.
Case 5: Minimization of emission
Electrical power generation from conventional sources of
energy emits hazardous gases into the environment. The
quantity of sulfur oxides SOx and nitrogen oxides NOx
emission in tones per hr (t/h) is higher with rise in generated
power (in p.u) next the relationship presented in equation
(27).The objective of OPF is to minimize emission :
( ) ( ) ( )2
1
, 0.001i Gi
i i
NBP
i i G i G i
i
f x u Emission P P e
=
= = + + +
(27)
where, 𝛼𝑖 , 𝛽𝑖 , 𝛾𝑖 , 𝜔𝑖 and 𝜇𝑖 are all emission coefficients
provided in [20]
The results yielded after optimization applied the CSHS
technique are presented in Table 5 and the trend of
optimization is shown in Fig.5a and 5b. The results appear
that the emission has been decreased from (0.3662 ton/h) to
(0.20476 ton/h), Thus, the results obtained are compared with
other techniques as shown in table 6.
Case 6: Minimization of real power loss
In this case, the purpose of the OPF problem is to
minimize power losses; the real power loss to be minimized
is formulated as follows:
( ) ( )2 2
1 1,
, 2 cosnl nl
loss ij i j i j ij
i j j i
f x u P G V V V V = =
= = + − (28)
where, 𝛿𝑖𝑗 = 𝛿𝑖 − 𝛿𝑗 is the difference in voltage angles
between bus 𝑖 and bus 𝑗 and 𝐺𝑖𝑗 is transfer conductance.
Figure 6. Objective function curve for CASE 5
Table 5. Optimum control variables for case 5 and case 6
Case 5 Case 6
Control variable CS-HS CS HS CS-HS CS HS
PG1 (MW) 63.5637 64.1721 64.5007 51.6524 51.6568 51.6718
PG2 (MW) 67.8700 67.3812 67.8834 79.6278 79.7125 80.0000
PG5 (MW) 50.0000 49.9997 50.0000 50.0000 49.9973 50.0000
PG8 (MW) 35.0000 35.0000 35.0000 35.0000 34.9675 35.0000
PG11 (MW) 30.0000 30.0000 30.0000 30.0000 29.9796 30.0000
PG13 (MW) 40.0000 40.0000 40.0000 40.0000 40.0000 40.0000
V1(p.u) 1.1000 1.1000 1.1000 1.1000 1.1000 1.1000
V2(p.u) 1.0933 1.0983 1.0781 1.0988 1.0972 1.1000
V5(p.u) 1.0740 1.0835 1.0440 1.0809 1.0791 1.0814
V8(p.u) 1.0854 1.0865 1.0384 1.0897 1.0862 1.0899
V11(p.u) 1.1000 1.0681 1.1000 1.0987 1.1000 0.9000
V13(p.u) 1.0965 1.0899 1.0392 1.0993 1.0993 1.1000
Qc10(Mvar) 0.2013 0.2011 5.0000 0.2025 1.2314 5.0000
Qc12(Mvar) 4.7013 2.1988 0 1.5592 4.4911 0
Qc15(Mvar) 2.4433 0.0063 5.0000 4.6077 0.1175 5.0000
Qc17(Mvar) 4.9988 0.0482 0 1.8248 4.5130 0
Qc20(Mvar) 5.0000 3.7074 5.0000 4.2253 0.0156 0
Qc21(Mvar) 0.4902 0.0803 5.0000 4.9905 3.0702 5.0000
Qc23(Mvar) 0.0401 2.4641 5.0000 0.2170 2.0676
5.0000
1
1ji
NGi
j LG
i j
VL H
V=
= −1,2,...,j NL=
busY
maxL
maxL
( ) ( )max
2
max
1
,i i
NG
i i G i G L
i
J x u a b P c P L=
= + + +
maxL
maxL
0 50 100 150 200 250 300 350 400 450 5000.2
0.205
0.21
0.215
0.22
0.225
0.23
0.235
0.24
0.245
Iteration
Em
issi
on (
t/h
)
CS
HS
CS-HS
184
Qc24(Mvar) 5.0000 2.5247 0 4.7142 3.4275 0
Qc29(Mvar) 3.7580 0 0 0.2424 3.2515 5.0000
T6–9 1.0161 1.0407 0.9000 1.0485 1.0602
1.0369
T6–10 0.9503 0.9326 1.1000 0.9000 0.9026
0.9000
T4–12 1.0136 1.0501 0.9000 0.9935 1.0052 1.1000
T28–27 0.9890 0.9844 0.9000 0.9743 0.9851 1.0327
Fuel cost ($/h) 944.3786 943.8722 946.7579 966.3352 966.4940
968.0725
VD 1.7171 1.2572 1.5023 1.8648 1.7185
0.7630
maxL 0.1294 0.1333 0.1268 0.1276 0.1288 0.1416
Emission (ton/h) 0.20476 0.2048 0.20506 0.2071 0.2071
0.2073
( )lossp MW 3.0337 3.1530 3.9841 2.8803 2.9137 3.2718
Table 6. Comparison of results for case 5 and case 6
Case 5 Case 6
Algorithms Emission (ton/h) Algorithms ( )lossp MW
CS-HS 0.20476 CS-HS 2.8803
CS 0.2048 CS 2.9137
HS 0.20506 HS 3.2
MSA [29] 0.20482 MSA [29] 3.1005
ARCBBO [27] 0.2048 ARCBBO [27] 3.1009
GBICA [30] 0.2049 GWO [31] 3.41
Figure 7. Objective function curve for CASE6
The tendency to decrease the objective function of total
real power transmission loss using the CSHS technique
appears Fig. 6. The optimal settings of the control variables
are presented in Table 5. In this case 6 by CSHS result in real
power losses of 2.8803MW, better than all the results
summarized in the table 6.
6.2 IEEE 57-bus test system
In order to exam the scalability of the suggested CSHS
technique, a greater test system is taken into account in this
article, which is the IEEE 57-bus test system. General system
data of 57-bus system are given in [33].
Case 7: Minimization of fuel cost
The goal of this case is to minimize the total generating
fuel cost. Hence, the objective function of this case is
presented by (18). The CSHS is run so as to find the optimal
settings for this case and the gained results are presented in
Table 7. The cost yielded for case 7 is (41662.1893$/h).
Table 7. Optimum control variables for case 7 and case 8
Case 7 Case 8
Control variable CSHS CS HS CSHS CS HS
PG1 (MW) 143.4303 144.8666 146.1972 141.7495 144.4274 149.8891
PG2 (MW) 91.8816 98.2138 100.0000 93.7131 92.3616 30.0000
PG3 (MW) 44.1534 46.1776 40.0000 45.7062 49.7799 46.8293
PG6 (MW) 75.3572 92.8569 30.0000 78.4672 53.5758 100.0000
PG8 (MW) 454.2645 470.7293 481.7175 457.3292 468.2286 469.4624
PG9 (MW) 96.6199 47.3525 100.0000 88.3416 93.9470 100.0000
PG12 (MW) 359.6901 366.0283 369.1071 361.2573 364.2777 370.5929
V1(p.u) 1.0661 1.0726 1.0737 1.0231 1.0211 1.1000
0 50 100 150 200 250 300 350 400 450 5002
3
4
5
6
7
8
9
10
11
Iteration
Plo
ss (
MW
)
CS
HS
CS-HS
185
V2(p.u) 1.0678 1.0764 1.0791 1.0275 1.0240 1.1000
V3(p.u) 1.0592 1.0645 1.0619 1.0160 1.0120 1.1000
V6(p.u) 1.0636 1.0651 1.0650 1.0257 1.0167 1.1000
V8(p.u) 1.0750 1.0769 1.1000 1.0457 1.0439 1.1000
V9(p.u) 1.0685 1.0675 1.0820 1.0282 1.0265 1.1000
V12(p.u) 1.0535 1.0572 1.0610 1.0072 1.0068 1.1000
Qc18(Mvar) 16.1246 0.1068 0 2.2705 0.3491 0
Qc25(Mvar) 15.2245 7.5240 16.5705 9.6667 6.8679 20.0000
Qc53(Mvar) 12.9365 6.7123 13.2102 16.1261 6.3440 20.0000
T4–18 1.1000 0.9063 1.1000 0.9652 0.9303 1.1000
T4–18 0.9941 1.0575 1.1000 0.9981 0.9886 1.0079
T21–20 1.0041 1.0133 0.9000 0.9790 0.9882 0.9839
T24–25 1.0188 0.9016 1.1000 0.9781 0.9576 1.1000
T24–25 0.9954 1.0332 0.9000 0.9511 0.9586 1.0002
T24–26 1.0166 1.0179 0.9931 1.0136 1.0258 1.0019
T7–29 1.0083 0.9943 1.0164 0.9985 0.9757 1.1000
T34–32 0.9631 0.9398 0.9000 0.9399 0.9344 0.9235
T11–41 0.9102 0.9547 0.9000 0.9000 0.9000 0.9000
T15–45 0.9843 0.9884 1.0017 0.9475 0.9525 1.0326
T14–46 0.9818 0.9854 1.0072 0.9534 0.9503 1.0330
T10–51 0.9934 0.9916 1.1000 0.9842 0.9767 1.0716
T13–49 0.9530 0.9519 0.9700 0.9218 0.9215 0.9939
T11–43 1.0047 0.9853 1.0992 0.9447 0.9429 1.1000
T40–56 0.9826 0.9640 0.9000 1.0448 1.0297 0.9000
T39–57 1.0279 0.9418 0.9000 0.9399 0.9467 1.1000
T9–55 1.0268 1.0084 1.0259 1.0180 0.9961 1.0704
Fuel cost ($/h) 41662.1893 41705.2382 41759.1345 41707.8925 41722.2262 41763.6339
VD 1.4928 1.5546 1.6527 0.6947 0.7111 1.4651
maxL 0.2807 0.2796 0.2889 0.2919 0.2914 0.2940
Emission (ton/h) 1.3331 1.4129 1.4620 1.3442 1.3893 1.4311
( )lossp MW 14.5973 15.4253 16.2232 15.7641 15.7980 15.9804
Table 8. Comparison of the results obtained for Case 7 and Case 8
Case 7 Case 8
Algorithms Fuel cost ($/h) Algorithms Fuel cost ($/h) VD (p.u)
CSHS 41662.1893 CSHS 41707.8925 0.6947
CS 41705.2382 CS 41722.2262 0.7111
HS 41759.1345 HS 41763.6339 1.4651
MSA [29] 41673.7231 MSA [29] 41714.9851 0.67818
ICBO [25] 41697.3324 FPA [29] 41726.3758 0.69723
Figure 8. Objective function curve for CASE 7
Case 8: Minimization of fuel cost and voltage deviation
The purpose of the objective function is to minimize
simultaneously both fuel cost and voltage deviation. The
converted single objective function next equation (21) with
weight factor w is chosen as 100, the results of such
optimization using the suggested CSHS technique are shown
in Table 7. This table shows that the VD has been decrease
from (1.4928 p.u.) to (0.6947 p.u.) compared with CASE 7.
Hence, the cost has slightly augmented from (41662.1893 $/h)
to (41707.8925 $/h) compared with CASE 7.
7. CONCLUSION
In this paper, a hybrid (CSHS) approach has been
proposed as a new solution to solve the OPF problem. At first,
in power systems, the OPF problem was reported as a non-
linear optimization problem with equality and inequality
constraints. Where several objective functions have been
considered to minimize the fuel cost, to improve the voltage
profile, and to enhance the voltage stability in normal
conditions. In addition, non-smooth cost objective function
0 500 1000 15004.16
4.18
4.2
4.22
4.24
4.26
4.28
4.3
4.32
4.34
4.36x 10
4 Convergence curve
Iteration
Fu
el c
ost
($/h
)
CS
HS
CS-HS
186
has been considered. The feasibility of the proposed CSHS
method for solving OPF problems is demonstrated by using
standard IEEE 30-bus and IEEE 57-bus test power systems.
The proposed method does always promise most optimal
solution and fast convergence. The simulation results
demonstrate the effectiveness and robustness of the proposed
algorithm to solve OPF problem in small and large test
systems. Furthermore, the proposed algorithms in this study
perform noticeably better than many other equivalent
optimization methods in finding solutions of OPF. Reduction
in hourly operation cost has been established almost in all the
cases studied under the scope of this literature.
REFERENCES
[1] Cain M, O’Neill R, Castillo A. (2012). History of
optimal power flow and formulations. FERC Staff Tech
Pap, pp. 1–36.
[2] Hinojosa VH, Araya R. (2013). Modeling a mixed-
integer-binary small-population evolutionary particle
swarm algorithm for solving the optimal power flow
problem in electric power systems. Appl. Soft Comput.
J 13: 3839–3852.
https://doi.org/10.1016/j.asoc.2013.05.005
[3] Ghasemi M, Ghavidel S, Akbari E, Vahed AA. (2014).
Solving non-linear, non-smooth and non-convex
optimal power flow problems using chaotic invasive
weed optimization algorithms based on chaos. Energy
73: 340–353.
https://doi.org/10.1016/j.energy.2014.06.026
[4] Abaali H, Lamchich MT, Raoufi M. (2007). Average
current mode to control the three phase shunt active
power filters under distorted and unbalanced Voltage
conditions. AMSE Journals, Series 2A 80(2): 68-81.
[5] Ambriz-Perez H, Acha E, Fuerte-Esquivel CR, De La
Torre A. (1998). Incorporation of a UPFC model in an
optimal power flow using Newton’s method. IEEE Proc
Gener Transm Distrib 145: 336e44.
https://doi.org/10.1049/ip-gtd:19981944
[6] Yan X, Quintana VH. (1999). Improving an interior-
point-based OPF by dynamic adjustments of step sizes
and tolerances. IEEE Trans. Power Syst 14: 709–717.
https://doi.org/10.1109/59.761902
[7] Al-Muhawesh TA, Qamber IS. (2008). The established
megawatt linear programming-based optimal power
flow model applied to the real power 56-bus system in
eastern province of Saudi Arabia. Energy 33: 12–21.
[8] Frank S, Steponavice I. (2012). Optimal power flow: A
bibliographic survey I. Formulations and Deterministic
Methods 3(3): 221–58.
[9] Chettih S, Khiat M, Chaker A. (2009). Var-voltage
control by particle swarm optimization (PSO) method-
application in the western algerian transmission system.
AMSE Journals, Series Modeling A 82(2): 65-79.
[10] Ben Attous D, Labb Y. (2010). Particle swarm
optimisation based optimal power flow for units with
non-smooth fuel cost functions. AMSE Journals, Series
Modelling A 83(3): 24-37.
[11] Benhamida F, Bendaoud A. (2009). A new formulation
of dynamic economic dispatch using a hopfield neural
network. AMSE Journals, Series Modelling A 82(2):
33-47.
[12] Bentouati B, Chettih S, El Sehiemy RA, Wang GG.
(2017). Elephant herding optimization for solving non-
convex optimal power flow problem. Journal of
Electrical and Electronics Engineering 10(1): 1-6.
[13] Bentouati B et al. (2016). Optimal power flow using the
moth flam optimizer: A case study of the algerian
power system. TELKOMINIKA (1): 3.
http://doi.org/10.11591/ijeecs.v1.i3.pp431-445
[14] Bhattacharya A, Chattopadhyay P. (2011). Application
of bio-geography-based optimization to solve different
optimal power flow problems. IET Gener Transm
Distrib 5(1): 70. https://doi.org/10.1049/iet-
gtd.2010.0237
[15] Abaci K, Yamacli V. (2016). Differential search
algorithm for solving multi-objective optimal power
flow problem. Int. J. Electr. Power Energy Syst. 79: 1–
10. https://doi.org/10.1016/j.ijepes.2015.12.021
[16] Bentouati B, et al. (2016). A solution to the optimal
power flow using multi-verse optimizer. J. Electrical
Systems 12-4 pp. 716-733,
[17] Roy PK, Paul C. (2015). Optimal power flow using krill
herd algorithm. Int. Trans. Electr. Energy Syst 25(8):
1397–1419. https://doi.org/10.1002/etep.1888
[18] Deb XYS. (2013). Cuckoo search: recent advances and
applications. https://doi.org/10.1007/s00521-013-1367-
1
[19] Geem ZW, Kim JH, Logan than GV. (2001). A new
heuristic optimization algorithm: Harmony search.
Simulation 76(2): 60–68.
https://doi.org/10.1177/003754970107600201
[20] Chaib AE, Bouchekara HREH, Mehasni R, Abido MA.
(2016). Optimal power flow with emission and non-
smooth cost functions using backtracking search
optimization algorithm. International Journal of
Electrical Power & Energy Systems 81: 64-77.
[21] Abou El Ela AA, Abido MA. (2010). Optimal power
flow using differential evolution algorithm. Electr.
Power Syst. Res 80 (7): 878–885.
[22] Duman S. (2016). Symbiotic organisms search
algorithm for optimal power flow problem based on
valve-point effect and prohibited zones. Neural Comput.
Appl. 28(11): 3571-3585.
[23] Niknam T, Narimani MR, Jabbari M, Malekpour AR.
(2011). A modified shuffle frog leaping algorithm for
multi-objective optimal power flow. Energy 36(11):
6420–32. https://doi.org/10.1016/j.energy.2011.09.027
[24] Niknam T, Narimani MR, Azizipanah-Abarghooee R.
(2012). A new hybrid algorithm for optimal power flow
considering prohibited zones and valve point effect.
Energy Convers. Manage 58: 197–206.
[25] Bouchekara HREH, Chaib AE, Abido MA, El-Sehiemy
RA. (2016). Optimal power flow using an Improved
Colliding Bodies Optimization algorithm. Applied Soft
Computing 42: 119-131.
[26] Mahdad B, Srairi K. (2016). Security constrained
optimal power flow solution using new adaptive
partitioning flower pollination algorithm. Applied Soft
Computing 46: 501-522.
[27] Ramesh Kumar A, Premalatha L. (2015). Optimal
power flow for a deregulated power system using
adaptive real coded biogeography-based optimization.
Electrical Power and Energy Systems 73: 393–399.
https://doi.org/10.1016/j.ijepes.2015.05.011
187
[28] Roy R, Jadhav HT. (2015). Optimal power flow
solution of power system incorporating stochastic wind
power using Gbest guided artificial bee colony
algorithm. Electrical Power and Energy Systems 64:
562–578.
[29] Mohamed AAA, Mohamed YS, El-Gaafary AAM,
Hemeida AM. (2017). Optimal power flow using moth
swarm algorithm. Electric Power Systems Research 142:
190–206.
[30] Ghasemi M, Ghavidel S, Ghanbarian M. (2015). Multi-
objective optimal electric power planning in the power
system using Gaussian bare-bones imperialist
competitive algorithm. Information Sciences. 294: 286-
304. https://doi.org/10.1016/j.ins.2014.09.051
[31] El-Fergany AA, Hasanien HM. (2015). Single and
multi-objective optimal power flow using grey wolf
optimizer and differential evolution algorithms. Electric
Power Components and Systems 43: 1548–1559.
https://doi.org/10.1080/15325008.2015.1041625
[32] Kessel P, Glavitsch H. (1986). Estimating the voltage
stability of a power system. IEEE Trans Power Deliv 1:
346–54. https://doi.org/10.1109/TPWRD.1986.4308013
[33] Zimmerman RD, Murillo-Sánchez CE, Thomas RJ.
Matpower http://www.pserc.cornell.edu/matpower
NOMENCLATURE
J (𝑥, 𝑢) Objective function.
h (x, u) Set of equality constraints.
g(x, u) Set of inequality constraints.
X State variables’ Vector.
U Control variables’ Vector.
PG Active power bus generator.
VG Voltage magnitude at 𝑖-th PV bus (generator
bus).
T Transformer tap setting.
QC Shunt VAR compensation.
PG1 Generator active power at slack bus.
VL Bus voltage of 𝑝-th load bus (PQ bus).
QG Reactive power generation of all generator
units.
SL Transmission line loading (or line flow).
NL and nl Number of load buses and the number of
transmission lines.
NC, NT
and NG
Number of VAR compensators, the number
of regulating transformers and the number of
generators respectively.
DP and
DQ
Active and reactive load demands.
ijG
Transfer conductance
ijB
Susceptance between bus 𝑖 and bus 𝑗, respectively.
min
pLV and
max
pLV
Lowest and upper load voltage of ith unit.
qlS
Apparent power flow of ith branch.
max
qlS
Maximum apparent power flow limit of ith
branch.
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